Yunchuan Yin | Searcharxiv

arXiv2015

W-graphs for Hecke algebras with unequal parameters(II)

This paper is the continuation of the work in~\cite{Yin}. In that paper we generalized the definition of $W$-graph ideal in the weighted Coxeter groups, and showed how to construct a $W$-graph from a given $W$-graph ideal in the case of unequal parameters. In this paper we study the full $W$-graphs for a given $W$-graph ideal. We show that there exist a pair of dual modules associated with a given $W$-graph ideal, they are connected by a duality map. % and the dual $W$-graph bases can be established. For each of the dual modules, the associated full $W$-graphs can be constructed.% among them, another pair of dual bases are obtained by using %the inversions of the relative Kazhdan-Lusztig polynomials. Our construction closely parallels that of Kazhdan and Lusztig~\cite{KL, Lusztig1, Lusztig2}, which can be regarded as the special case $J=\emptyset$. It also generalizes the work of Couillens~\cite{C}, Deodhar~\cite{Deodhar, Deodhar2}, and Douglass \cite{MD}, corresponding to the parabolic cases.

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arXiv2010

Kazhdan-Lusztig basis for generic Specht modules

In this paper, we let $\Hecke$ be the Hecke algebra associated with a finite Coxeter group $W$ and with one-parameter, over the ring of scalars $\Alg=\mathbb{Z}(q, q^{-1})$. With an elementary method, we introduce a cellular basis of $\Hecke$ indexed by the sets $E_J (J\subseteq S)$ and obtain a general theory of "Specht modules". We provide an algorithm for $W\!$-graphs for the "generic Specht module", which associates with the Kazhdan and Lusztig cell ( or more generally, a union of cells of $W$ ) containing the longest element of a parabolic subgroup $W_J$ for appropriate $J\subseteq S$. As an example of applications, we show a construction of $W\!$-graphs for the Hecke algebra of type $A$.

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